Dave,
You appear to have chosen a French helicopter (with the blades rotating clockwise from above), but no matter either way.
I think the answer is that both systems have a phase lag of 90 degrees. A blade will not move instantly to a new position, whatever force causes the motion. For the aircraft to roll, the cyclic pitch increase would need to be put in as the rotor blades pass the 6 o'clock position of the aircraft.
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With regard to the centripetal force rather than gyroscopic force; you may be more correct. (I did originally include the term but I edited the post many times until my eyes were going crossed).
My understanding is that both forces are a function of the same phenomena. Newton stated that a body tends to move in a straight line unless acted upon by another force. A body / mass forced to rotate around an axis (because it is fixed to it), instead of continuing in its preferred straight line, can do so only by experiencing a force, in this case imparted by the attachment. Newton also stated that every reaction imparts an equal and opposite reaction and so the axis by definition must be experiencing an equal force to the rotating mass. Or vice versa.
The characteristics of gyroscopes can be explained by those two Newton's laws (I'll be very honest here and admit that I can't currently remember which he quoted first second or third!).
Precession is simply a description of the movement of a rotating body out of its plane of rotation by a displacing force acting normal (90 degrees / perpendicular) to the ORIGINAL plane of rotation. The force acts, the rotating body moves in the direction of the force (whilst still rotating around the axis) until the displacing force can no longer act because the plane of rotation has changed so that it is in the same direction as the displacing force; i.e. through 90 degrees but 90 degrees further round the "disk" than where the force was applied.
I probably need a model or a diagram to put that across properly.
ShyT